The Stacks project

Lemma 10.11.3. Let $R$ be a ring and let $M$ be an $R$-module. Then $M$ is the colimit of a directed system $(M_ i, \mu _{ij})$ of $R$-modules with all $M_ i$ finitely presented $R$-modules.

Proof. Consider any finite subset $S \subset M$ and any finite collection of relations $E$ among the elements of $S$. So each $s \in S$ corresponds to $x_ s \in M$ and each $e \in E$ consists of a vector of elements $f_{e, s} \in R$ such that $\sum f_{e, s} x_ s = 0$. Let $M_{S, E}$ be the cokernel of the map

\[ R^{\# E} \longrightarrow R^{\# S}, \quad (g_ e)_{e\in E} \longmapsto (\sum g_ e f_{e, s})_{s\in S}. \]

There are canonical maps $M_{S, E} \to M$. If $S \subset S'$ and if the elements of $E$ correspond, via this map, to relations in $E'$, then there is an obvious map $M_{S, E} \to M_{S', E'}$ commuting with the maps to $M$. Let $I$ be the set of pairs $(S, E)$ with ordering by inclusion as above. It is clear that the colimit of this directed system is $M$. $\square$

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